Every complete doubling metric space carries a doubling measure

Luukkainen, Jouni; Saksman, Eero

doi:10.1090/S0002-9939-98-04201-4

Every complete doubling metric space carries a doubling measure
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by Jouni Luukkainen and Eero Saksman PDF

Proc. Amer. Math. Soc. 126 (1998), 531-534 Request permission

Abstract:

We prove that a complete metric space $X$ carries a doubling measure if and only if $X$ is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on $X$ and of the homogeneity exponents of $X$ are equal. We also show that a closed subset $X$ of $\mathbf {R}^{n}$ carries a measure of homogeneity exponent $n$. These results are based on the case of compact $X$ due to Vol$^{\prime }$berg and Konyagin.

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